Our theory builds on the work of [1]} who show consistency of a cross-validation scheme in their Theorems 1 and 2. However inspection of their proofs shows that their conclusions as stated may need some caveats. Firstly, the proofs require all change-points to occur at even locations.
This simplification of the model may be justified when the noise and signal strength are bounded away from 0 and \(\infty \)  respectively. However by bypassing the issue of poor predictive performance described above, the insights of this asymptotic regime for finite samples are thus perhaps somewhat limited.
Secondly, it is unclear to us how the arguments in their proofs may be extended to allow for
the maximum number of change-points considered and the true number of change-points to diverge; see the discussion before and after (REF ) in Section  in the supplementary material of our paper. This seems particularly relevant given that the number of change-points is the object of inference.
In contrast, our result (Theorem REF ) allows for the number of change-points to tend to infinity. We verify the conditions of our general result for the case of optimal partitioning by employing a new result (Theorem REF  in Section REF ) on the existence of estimated change-points in the neighbourhood of true changes even when the number of changes has been incorrectly specified, which may be of independent interest.
